On C-infinity well-posedness of hyperbolic systems with multiplicities

In this paper we study first order hyperbolic systems of any order with multiple characteristics (weakly hyperbolic) and time-dependent analytic co- efficients. The main question is when the Cauchy problem for such systems is well-posed in C ∞ and in D 0 . We prove that the analyticity of the coefficients com- bined with suitable hypotheses on the eigenvalues guarantee the C ∞ well-posedness of the corresponding Cauchy problem